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Mathematics · Ch 7 — Integrals

Integration by Substitution

7.3.1

Integration by Substitution

The Method of Substitution

Integration by substitution reverses the chain rule. It transforms a complicated integral into a simpler one by changing the variable of integration. For ∫f(x) dx\int f(x) \, dx, set x=g(t)x = g(t) where gg is differentiable; then:

dxdt=g′(t)sodx=g′(t) dt\frac{dx}{dt} = g'(t) \quad \text{so} \quad dx = g'(t) \, dt

giving the substitution formula:

∫f(x) dx=∫f(g(t)) g′(t) dt\int f(x) \, dx = \int f(g(t)) \, g'(t) \, dt

The art lies in choosing the right substitution — usually a function whose derivative also appears in the integrand, so that its derivative gets absorbed into dtdt.

Tip

Choosing the substitution

Look for a function uu such that its derivative dudu appears as a factor in the integrand. Set u=that functionu = \text{that function}, then du=its derivative×dxdu = \text{its derivative} \times dx, and the integral simplifies.

Standard Integrals of Trigonometric Functions

These four integrals are derived using substitution and are important enough to be used directly in future work.

Integral of tan⁡x\tan x

∫tan⁡x dx=∫sin⁡xcos⁡x dx\int \tan x \, dx = \int \frac{\sin x}{\cos x} \, dx

Substitute cos⁡x=t\cos x = t, so −sin⁡x dx=dt-\sin x \, dx = dt and sin⁡x dx=−dt\sin x \, dx = -dt:

∫sin⁡xcos⁡x dx=∫−dtt=−log⁡∣t∣+C=−log⁡∣cos⁡x∣+C=log⁡∣sec⁡x∣+C\int \frac{\sin x}{\cos x} \, dx = \int \frac{-dt}{t} = -\log |t| + C = -\log |\cos x| + C = \log |\sec x| + C

∫tan⁡x dx=log⁡∣sec⁡x∣+C\int \tan x \, dx = \log |\sec x| + C

Integral of cot⁡x\cot x

∫cot⁡x dx=∫cos⁡xsin⁡x dx\int \cot x \, dx = \int \frac{\cos x}{\sin x} \, dx

Substitute sin⁡x=t\sin x = t, so cos⁡x dx=dt\cos x \, dx = dt:

∫cos⁡xsin⁡x dx=∫dtt=log⁡∣t∣+C=log⁡∣sin⁡x∣+C\int \frac{\cos x}{\sin x} \, dx = \int \frac{dt}{t} = \log |t| + C = \log |\sin x| + C

∫cot⁡x dx=log⁡∣sin⁡x∣+C\int \cot x \, dx = \log |\sin x| + C

Integral of sec⁡x\sec x

Multiply numerator and denominator by (sec⁡x+tan⁡x)(\sec x + \tan x):

∫sec⁡x dx=∫sec⁡x(sec⁡x+tan⁡x)sec⁡x+tan⁡x dx\int \sec x \, dx = \int \frac{\sec x (\sec x + \tan x)}{\sec x + \tan x} \, dx

Substitute sec⁡x+tan⁡x=t\sec x + \tan x = t. The derivative sec⁡xtan⁡x+sec⁡2x=sec⁡x(tan⁡x+sec⁡x)\sec x \tan x + \sec^2 x = \sec x (\tan x + \sec x) is exactly the numerator, so dt=sec⁡x(sec⁡x+tan⁡x) dxdt = \sec x (\sec x + \tan x) \, dx:

=∫dtt=log⁡∣t∣+C=log⁡∣sec⁡x+tan⁡x∣+C= \int \frac{dt}{t} = \log |t| + C = \log |\sec x + \tan x| + C

∫sec⁡x dx=log⁡∣sec⁡x+tan⁡x∣+C\int \sec x \, dx = \log |\sec x + \tan x| + C

Integral of csc⁡x\csc x …

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